Quantum State Vectors: Algebraic Operations
The binary system that is used in digital devices has two states: 0 and 1. In quantum computing, these states are represented as vectors; but unlike normal vectors, state vectors are denoted using the ket notation, |v⟩. Thus, 0 = |0⟩ and 1 = |1⟩. Quantum state vectors simply point to a specific point in space that corresponds to a particular quantum state.
A vector space, V, over a field, F, is a set of vectors where two conditions hold:
1. Adding two vectors that are elements of the vector space, V, will yield a third vector, which is also an element of V.
2. Scalar multiplication between a vector,| a⟩ ∈ V, and a scalar, n ∈ F, is also a member of V.
Algebraic Properties of Vectors
State vectors can be expressed in the [x,y] format such that:
Recap:
Vector Addition — The sum of two vectors is another vector.
Given vectors |α⟩and|β⟩: |α⟩+|β⟩ = |γ. In normal vector operations:
Additive Properties of Vectors
1. Vector addition is commutative. |α⟩+|β⟩ = |β⟩ +|α⟩
2. Vector addition is associative. |α⟩+(|β⟩+|γ⟩)= (|α⟩ +|β⟩)+|γ⟩
The null/zero vector, |0⟩, has two properties that are shown in equations 1 and 2 below.
Vector Multiplication — vector multiplication can be done in three ways: scalar, dot, or cross product.
Scalar Product involves the multiplication of a vector by some real number m or n.
1. Commutative Law of Multiplication
2. Associative Law of Multiplication
2. Distributive Law of Multiplication
Matrix Multiplication
Matrices are mathematical objects that transform vectors to other vectors:
| v⟩ → | v′⟩ = M |v⟩. Matrix multiplication is given by the general formula:
Vector multiplication can also be an inner/dot product or a cross product.
Inner/Dot Product
In normal vector operations, the dot product of two vectors, A and B, is defined as the product of the magnitudes of A and B and the cosine of the angle θ between them. Therefore, A.B = AB cos θ and A.B is a scalar.
However, the dot product of two state vectors |α⟩and|β⟩ is a complex number:
The dot product has the following properties:
1. Commutative
2. The dot product of a vector by itslef is greater than or equal to 0 and |α⟩.|α⟩ is only equal to zero if |α⟩ is 0.
3. Associative
Cross Product
The cross product of A and B is a vector C = AxB (read A cross B), which is the product of the magnitudes of A and B and the sine of the angle θ between them.
AxB = AB sin θ u where is a unit vector indicating the direction of A B.
Properties of the cross product will be discussed in a different article.
Going forward, these algebraic properties will be essential in discussing the characteristics of state vectors and quantum gates.
